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The classical Yang-Mills equation in the presence of a source $J^\nu(x)$ can be written as $$ \partial_\mu F^{\mu \nu} - i g [A_\mu, F^{\mu \nu}] = J^\nu (x), $$ where $F^{\mu \nu} = \partial^\mu A^\nu(x) - \partial^\nu A^\mu(x) -ig [A^\mu(x),A^\nu(x)]$, and $A^\mu(x) = t^a A_a^\mu(x)$ with $t^a$ as the generators of the gauge group. This equation can be further simplified to express it in terms of $A_a$.

My question is: Given an expression for the current $J^\mu(x)$, is it possible to solve this equation numerically for $A^\mu(x)$? In other words, can we solve this system of four, non-linear, second-order differential equations?

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Usually it is very difficult to solve nonlinear 2nd order differential equations of several variables. But in general relativity it is attempted in the so called Numerical General Relativity (non-quantum).

The approach for Yang-Mills (Y-M) theories is a different one. One computes Feynman's path integral for the corresponding Y-M theory on a lattice. The approach is called Lattice Gauge Theory. The good thing about the path integral approach is that it considers the quantum aspect of the theory and secondly that the path integral is well defined on a lattice.

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  • $\begingroup$ Thanks for the answer, it was very helpful. I am aware of the existence of Lattice QCD but, as far as I know, in lattice calculations the degrees of freedom are the Wilson lines rather than the gauge field itself. On the other hand I am interested in the classical limit of the theory so that the path integral reduces to its saddle point solution. $\endgroup$
    – aruera
    Feb 7 at 13:20
  • $\begingroup$ Well I cannot tell you about that. $\endgroup$ Feb 7 at 15:38

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